Theorem pmtrsn 19590

Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)

Assertion

Ref	Expression
pmtrsn	⊢ (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn

Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5412	. . 3 ⊢ {𝐴} ∈ V
2		eqid 2763	. . . 4 ⊢ (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
3	2	pmtrfval 19521	. . 3 ⊢ ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
4	1, 3	ax-mp 5	. 2 ⊢ (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
5		eqid 2763	. . . . 5 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	5	dmmpt 6243	. . . 4 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7		2on0 8469	. . . . . . . . 9 ⊢ 2o ≠ ∅
8		ensymb 9000	. . . . . . . . . 10 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅)
9		en0 9016	. . . . . . . . . 10 ⊢ (2o ≈ ∅ ↔ 2o = ∅)
10	8, 9	bitri 278	. . . . . . . . 9 ⊢ (∅ ≈ 2o ↔ 2o = ∅)
11	7, 10	nemtbir 3054	. . . . . . . 8 ⊢ ¬ ∅ ≈ 2o
12		snnen2o 9206	. . . . . . . 8 ⊢ ¬ {𝐴} ≈ 2o
13		0ex 5271	. . . . . . . . 9 ⊢ ∅ ∈ V
14		breq1 5113	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ≈ 2o ↔ ∅ ≈ 2o))
15	14	notbid 321	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (¬ 𝑦 ≈ 2o ↔ ¬ ∅ ≈ 2o))
16		breq1 5113	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝑦 ≈ 2o ↔ {𝐴} ≈ 2o))
17	16	notbid 321	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2o ↔ ¬ {𝐴} ≈ 2o))
18	13, 1, 15, 17	ralpr 4667	. . . . . . . 8 ⊢ (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o ↔ (¬ ∅ ≈ 2o ∧ ¬ {𝐴} ≈ 2o))
19	11, 12, 18	mpbir2an 723	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o
20		pwsn 4866	. . . . . . . 8 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
21	20	raleqi 3321	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o)
22	19, 21	mpbir 234	. . . . . 6 ⊢ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o
23		rabeq0 4346	. . . . . 6 ⊢ ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o)
24	22, 23	mpbir 234	. . . . 5 ⊢ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅
25	24	rabeqi 3429	. . . 4 ⊢ {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
26		rab0 4343	. . . 4 ⊢ {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
27	6, 25, 26	3eqtri 2790	. . 3 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
28		mptrel 5814	. . . 4 ⊢ Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29		reldm0 5920	. . . 4 ⊢ (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
30	28, 29	ax-mp 5	. . 3 ⊢ ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
31	27, 30	mpbir 234	. 2 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
32	4, 31	eqtri 2786	1 ⊢ (pmTrsp‘{𝐴}) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ∖ cdif 3903  ∅c0 4287  ifcif 4488  𝒫 cpw 4563  {csn 4590  {cpr 4592  ∪ cuni 4873   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5663  Rel wrel 5668  ‘cfv 6538  2_oc2o 8448   ≈ cen 8941  pmTrspcpmtr 19512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-1o 8454  df-2o 8455  df-er 8695  df-en 8945  df-pmtr 19513

This theorem is referenced by:  psgnsn  19591